Vector Addition and Scalar Multiplication

Vectors are mathematical quantities used to represent concepts such as force or velocity which have both a magnitude and a direction.

The figure below shows vector v with initial point A and terminal point B.

Components of a Vector

The component form of vector v with initial point A(a1,a2) and terminal point B(b1,b2) is given by

v = < b1 - a1 , b2 - a2 >

If a vector is given by v = < v1 , v2 > , it magnitude || v || is given by

|| v || = √(v1 2 + v2 2)

Example 1 : Find the components and the magnitude of vector v with initial point A(2,3) and terminal point B(4,5).

Solution to example 1:

Use above definition to find vector v

v = < v1 , v2 > = < b1 - a1 , b2 - a2 >

= < 4 - 2 , 5 - 3 > = < 2 , 2 >

and its magnitude || v ||

|| v || = √(v1 2 + v2 2)

= SQRT(2 2 + 2 2) = √(8) = 2 √(2)

Scalar Multiplication of a Vector

The scalar multiplication of vector v = < v1 , v2 > by a real number k is the vector k v given by

k v = < k v1 , k v2 >

Addition of two Vectors

The addition of two vectors v(v1 , v2) and u (u1 , u2) gives vector

v + u = < v1 + u1 , v2 + u2>

Below is an html5 applets that may be used to understand the geometrical explanation of the addition of two vectors. Enter components of vectors A and B and use buttons to draw, add, zoom in and out as well as translate the system of axes.